''Luchthaven'' sounds a bit more formal to me as a native speaker. It's the kind of term that an airport in the Netherlands would officially be referred to, as in ''Luchthaven Schiphol''. I think ''vliegveld'' is more common in casual daily speech, but ''luchthaven'' is not wrong either.

A" luchthaven" is what we call an "airport" this should therefore not have to be an international airport. A regional airport also is a "luchthaven". A "vliegveld", is more likely an "airfield", a place where smaller private aircrafts take off and land. Sometimes on these smaller airfields, the runway is just a flat grass or sandy area.